Almost totally complex points on elliptic curves

TL;DR

This paper extends Darmon's ATR method to construct algebraic points on elliptic curves over certain number fields, providing explicit examples and numerical evidence supporting their conjectural algebraicity.

Contribution

It introduces a new construction of Darmon-like points on elliptic curves over almost totally complex fields using higher-dimensional modular abelian varieties.

Findings

Constructed explicit Darmon-like points on elliptic curves.

Provided numerical evidence supporting algebraicity conjectures.

Extended ATR method to higher-dimensional cases.

Abstract

Let $F/F_0$ be a quadratic extension of totally real number fields, and let $E$ be an elliptic curve over $F$ which is isogenous to its Galois conjugate over $F_0$. A quadratic extension $M/F$ is said to be almost totally complex (ATC) if all archimedean places of $F$ but one extend to a complex place of $M$. The main goal of this note is to provide a new construction of a supply of Darmon-like points on $E$, which are conjecturally defined over certain ring class fields of $M$. These points are constructed by means of an extension of Darmon's ATR method to higher dimensional modular abelian varieties, from which they inherit the following features: they are algebraic provided Darmon's conjectures on ATR points hold true, and they are explicitly computable, as we illustrate with a detailed example that provides certain numerical evidence for the validity of our conjectures.